L7 · Polynomials clinic
“I can look at any polynomial problem from this unit, choose the right move - combine, subtract with a flip, expand, factor out a GCF, product-sum, difference of squares, or declare non-factorable - and carry it out with a check.”
Three problems, three different first moves. Problem 1: Simplify (2x^2 + 3x - 4) - (x^2 - 5x + 1). The - between brackets says: flip, then combine. Flip: -x^2 + 5x - 1. Combine: x^2 + 8x - 5. Problem 2: Expand (x + 2)(x^2 - x + 3). The x between brackets says: grid, 2 x 3 = six cells. Cells: x^3, -x^2, 3x, 2x^2, -2x, 6 - combine: x^3 + x^2 + x + 6. Problem 3: Factor 3x^2 + 12x + 9. "Rewrite as a product" says: factor - and the FIRST factoring move is always the GCF. GCF is 3: 3(x^2 + 4x + 3). Now the bracket is an x^2 + bx + c: pair product 3, sum 4 - (1, 3). Fully factored: 3(x + 1)(x + 3). Verify the inner factoring: (x + 1)(x + 3) = x^2 + 4x + 3. Say it plainly: the operation symbol in the problem is the instruction. Read it before touching a single term.
Why did Problem 3 need the GCF step BEFORE product-sum?
Factor 2x^2 - 8x + 6 completely. -> 2(x - 1)(x - 3) (GCF 2 out: 2(x^2 - 4x + 3); pair product 3, sum -4: -1 and -3. Check inner: (x - 1)(x - 3) = x^2 - 4x + 3)
"Expand (x - 4)(x^2 + 3x - 2): the six cells are x^3, 3x^2, -2x, ___, ___, +8." -> -4x^2 and -12x (the -4 row crossing x^2 and 3x; full result: x^3 - x^2 - 14x + 8)
10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.